The value of a probability density function is zero or positive for every value of x.
Correct Answer
True
True. Areas under the density are probabilities, which cannot be negative, so f(x) can never dip below the x-axis. It can be zero where X cannot occur.
Category: Probability as Area
Question 2
A probability density function can have a height greater than 1 at some values of x.
Correct Answer
True
True. Only the total area must equal 1, not the height. For example, X ~ U(0, 0.5) has height 1/0.5 = 2 across its whole range, and its area is 0.5 times 2 = 1.
Category: Probability as Area
Question 3
For X ~ U(0, 10), P(X < 3) equals 0.3.
Correct Answer
True
True. The area is the base times the height, (3 - 0)(1/10) = 0.3. For a uniform variable, each probability is the length of the interval divided by the full width b - a.
Category: Uniform Calculations
Question 4
An exponential random variable takes only values that are zero or positive.
Correct Answer
True
True. The exponential density f(x) = m e^(-mx) is defined for x >= 0, because it measures a waiting time or a lifetime, which cannot be negative.
Category: Exponential Distribution
Question 5
An exponential distribution with a mean of 5 minutes has a standard deviation of 5 minutes.
Correct Answer
True
True. For an exponential distribution, the standard deviation equals the mean, and both equal 1/m. Here m = 0.2, so mu = sigma = 5 minutes.
Category: Exponential Distribution
Question 6
If X ~ U(0, 20) and it is known that X > 10, then X is uniformly distributed between 10 and 20.
Correct Answer
True
True. Conditioning a uniform variable on a sub-interval shrinks the sample space, and the values that remain are still equally likely. The new density is 1/(20 - 10) = 1/10 on 10 to 20.
Category: Uniform Distribution
Question 7
A continuous random variable can take any value within an interval, not only whole numbers.
Correct Answer
True
True. Continuous variables such as time, height or weight are measured on a scale, so any value in a range is possible, including values with many decimal places. That is why probabilities come from areas rather than from a list of values.
Category: Continuous Variables
Question 8
The exponential distribution is skewed to the right.
Correct Answer
True
True. The exponential density is highest at x = 0 and decreases steadily, leaving a long tail to the right. Short waits are common and long waits are rare but possible.
Category: Exponential Distribution
Question 9
The probability density function of X ~ Exp(m) is f(x) = m e^(-mx) for x >= 0.
Correct Answer
True
True. This is the exponential density with decay parameter m. Its value at x = 0 is m, and it decreases toward 0 as x grows.
Category: Exponential Distribution
Question 10
For a uniform distribution on a to b, the mean and the median are equal.
Correct Answer
True
True. The uniform density is a rectangle, which is symmetric about its center. Both the mean and the median fall at the midpoint (a + b)/2.
Category: Uniform Distribution
Question 11
The exponential distribution is used to model the number of events that occur in a fixed time interval.
Correct Answer
False
False. Counts of events in a fixed interval follow a Poisson distribution, which is discrete. The exponential distribution models the continuous time between events or until the next event.
Category: Poisson Connection
Question 12
For an exponential distribution, the median is greater than the mean.
Correct Answer
False
False. The median of an exponential distribution is less than the mean. The median is ln(2)/m, about 0.693 times the mean, because the long right tail pulls the mean upward.
Category: Exponential Calculations
Question 13
In a uniform distribution, values near the mean are more likely than values near the endpoints.
Correct Answer
False
False. In a uniform distribution the density is the same height everywhere from a to b, so intervals of equal length have equal probability no matter where they sit. Values near the mean are no more likely than values near the endpoints.
Category: Uniform Distribution
Question 14
The time it takes a runner to finish a mile is a discrete random variable.
Correct Answer
False
False. Time is measured on a continuous scale and can take any value in a range, so it is a continuous random variable. Discrete variables take separate, countable values, such as the number of laps completed.
Category: Continuous Variables
Question 15
In an exponential distribution, a larger decay parameter m gives a larger mean.
Correct Answer
False
False. The mean is 1/m, so a larger decay parameter gives a smaller mean. A larger m makes the density drop off faster, so short waits become even more common.
Category: Exponential Distribution
Question 16
For X ~ Exp(m), P(X < x) = e^(-mx).
Correct Answer
False
False. The expression e^(-mx) gives the area to the right, P(X > x). The cumulative probability is P(X < x) = 1 - e^(-mx).
Category: Exponential Calculations
Question 17
The standard deviation of a uniform distribution on a to b is (b - a)/12.
Correct Answer
False
False. The variance of a uniform distribution is (b - a)^2/12, so the standard deviation is sqrt((b - a)^2/12), which equals (b - a)/sqrt(12). Dividing the width by 12 without the square and square root gives the wrong value.
Category: Uniform Calculations
Question 18
For a continuous random variable with density f(x), P(1 < X < 3) equals f(3) - f(1).
Correct Answer
False
False. Probabilities are areas under the density, not differences in its height. P(1 < X < 3) is the area under f(x) from 1 to 3, which equals the cumulative probability at 3 minus the cumulative probability at 1.
Category: Probability as Area
Question 19
The probability distribution of a continuous random variable is given by listing each possible value with its probability.
Correct Answer
False
False. A listed table of values and probabilities describes a discrete random variable. A continuous variable has infinitely many possible values, each with probability 0, so its distribution is described by a density curve whose areas give probabilities.
Category: Continuous Variables
Question 20
In an exponential distribution with decay parameter m = 0.5, the density at x = 0 equals 1.
Correct Answer
False
False. Substituting x = 0 into f(x) = m e^(-mx) gives f(0) = m, since e^0 = 1, so here the density at 0 is 0.5. The starting height of an exponential density always equals its decay parameter.
Category: Exponential Distribution